CMPE13 Cyrus Bazeghi Chapter 17 Recursion

CMPE13 What is Recursion? A recursive function is one that solves its task by calling itself on smaller pieces of data. –Similar to recurrence function in mathematics. –Like iteration -- can be used interchangeably; sometimes recursion results in a simpler solution.

CMPE13 Recursion Example Example: Running sum ( ) Mathematical Definition: RunningSum(1) = 1 RunningSum(n) = n + RunningSum(n-1) Recursive Function: int RunningSum(int n) { if (n == 1) return 1; else return n + RunningSum(n-1); }

CMPE Executing RunningSum RunningSum(4) RunningSum(3) RunningSum(2) RunningSum(1) return value = 1 return value = 3 return value = 6 return value = 10 return 1; return 2 + RunningSum(1); return 3 + RunningSum(2); return 4 + RunningSum(3); res = RunningSum(4);

CMPE High-Level Example: Binary Search Given a sorted set of exams, in alphabetical order, find the exam for a particular student. 1. Look at the exam halfway through the pile. 2. If it matches the name, we're done; if it does not match, then... 3a. If the name is greater (alphabetically), then search the upper half of the stack. 3b. If the name is less than the halfway point, then search the lower half of the stack.

CMPE Binary Search: Pseudocode Pseudocode is a way to describe algorithms without completely coding them in C. FindExam(studentName, start, end) { halfwayPoint = (end + start)/2; if (end < start) ExamNotFound(); /* exam not in stack */ else if (studentName == NameOfExam(halfwayPoint)) ExamFound(halfwayPoint); /* found exam! */ else if (studentName < NameOfExam(halfwayPoint)) /* search lower half */ FindExam(studentName, start, halfwayPoint - 1); else /* search upper half */ FindExam(studentName, halfwayPoint + 1, end); }

CMPE High-Level Example: Towers of Hanoi Task: Move all disks from current post to another post. Rules: (1) Can only move one disk at a time. (2) A larger disk can never be placed on top of a smaller disk. (3) May use third post for temporary storage. Post 1Post 2Post 3

CMPE Task Decomposition Suppose disks start on Post 1, and target is Post Move top n-1 disks to Post 2. How many moves? 2. Move largest disk to Post Move n-1 disks from Post 2 to Post

CMPE Task Decomposition (cont.) Task 1 is really the same problem, with fewer disks and a different target post. –"Move n-1 disks from Post 1 to Post 2." And Task 3 is also the same problem, with fewer disks and different starting and target posts. –"Move n-1 disks from Post 2 to Post 3." So this is a recursive algorithm. –The terminal case is moving the smallest disk -- can move directly without using third post. –Number disks from 1 (smallest) to n (largest).

CMPE Towers of Hanoi: Pseudocode MoveDisk(diskNumber, startPost, endPost, midPost) { if (diskNumber > 1) { /* Move top n-1 disks to mid post */ MoveDisk(diskNumber-1, startPost, midPost, endPost); printf("Move disk number %d from %d to %d.\n", diskNumber, startPost, endPost); /* Move n-1 disks from mid post to end post */ MoveDisk(diskNumber-1, midPost, endPost, startPost); } else printf("Move disk number 1 from %d to %d.\n", startPost, endPost); }

CMPE Detailed Example: Fibonacci Numbers Mathematical Definition: In other words, the n-th Fibonacci number is the sum of the previous two Fibonacci numbers.

CMPE Fibonacci: C Code int Fibonacci(int n) { if ((n == 0) || (n == 1)) return 1; else return Fibonacci(n-1) + Fibonacci(n-2); }

CMPE Activation Records Whenever Fibonacci is invoked, a new activation record is pushed onto the stack. Fib(1) R6 Fib(2) Fib(3) main main calls Fibonacci(3) Fibonacci(3) calls Fibonacci(2) Fibonacci(2) calls Fibonacci(1) R6 Fib(3) main R6 Fib(2) Fib(3) main

CMPE Activation Records (cont.) Fibonacci(1) returns, Fibonacci(2) calls Fibonacci(0) Fibonacci(2) returns, Fibonacci(3) calls Fibonacci(1) Fibonacci(3) returns R6 main R6 Fib(1) Fib(3) main Fib(0) R6 Fib(2) Fib(3) main

CMPE Tracing the Function Calls If we are debugging this program, we might want to trace all the calls of Fibonacci. Note: A trace will also contain the arguments passed into the function. For Fibonacci(3), a trace looks like: Fibonacci(3) Fibonacci(2) Fibonacci(1) Fibonacci(0) Fibonacci(1)

CMPE Fibonacci: LC-3 Code Activation Record temp dynamic link return address return value n bookkeeping arg Compiler generates temporary variable to hold result of first Fibonacci call. local

CMPE Recursively converts an unsigned integer as a string of ASCII characters. If integer <10, convert to char and print. Else, call self on first (n-1) digits and then print last digit. A Final C Example: Printing an Integer

CMPE A Final C Example: Printing an Integer void IntToAscii(int num) { int prefix, currDigit; if (num < 10) putchar(num + '0'); /* prints single char */ else { prefix = num / 10; /* shift right one digit */ IntToAscii(prefix); /* print shifted num */ /* then print shifted digit */ currDigit = num % 10; putchar(currDigit + '0'); } }

CMPE Trace of IntToAscii Calling IntToAscii with parameter 12345: IntToAscii(12345) IntToAscii(1234) IntToAscii(123) IntToAscii(12) IntToAscii(1) putchar('1') putchar('2') putchar('3') putchar('4') putchar('5')